where w(t), h(t), a(t) and are distortion functions (see below), and the star "*" is the convolution operator. Consequently, a generalized MR-spectrum takes a form:
where W(v):= F[w(t)], H(v):= F[h(t)], A(v):= F[a(t)] and N(v):= F[n(t)] are Fourier representations of the individual distortion functions. These distortion functions are interpreted as follows:
w(t), W(v) := inhomogeneity functions associated with a static and an rf-magnetic field Bo,B1;
h(t), H(v) := excitation functions associated with an rf-magnetic field B1;
a(t), A(v) := apodization functions associated with a data processing;
n(t), N(v) := random noise functions associated with a signal path.
A complex inhomogeneity function corresponds to a spatial inhomogeneity of the static magnetic field Bo, to a time instability of B1, to an inhomogeneity of the rf-field B1, and to an imperfect quadrature detection. Obviously, the inhomogeneity function does not influence the noise term n(t). (Analogically for the spectral co-domain.) For spin-echo experiments, this inhomogeneity function shows a periodic structure.
A complex excitation function H(v) describes both the amplitude and the phase modulations of the inhomogeneity-distorted spectrum Z(v,.). The function H(v) corresponds to a particular rf-excitation of a spin ensemble. Note that H(v) does not act upon the noise term N(v). This corresponds to a real physical situation where the noise is added to the signal after the inhomogeneity-, amplitude- and phase-distortions. In contrast to this physical situation, conventional amplitude and phase corrections act also upon the noise term N(v). (Analogically for the time-domain.)
A real apodization function a(t) may be used to describe various processing associated operations: "matched" filterings, signal truncations (one-sided or two-sided), non-equidistant samplings, etc. Note that the apodization acts also on the noise term n(t). (Analogically for the spectral co-domain.)
A real-time interactive MATLAB-simulator has been developed (see Fig. 1, 2) for modelling the complex inhomogeneity, noise, phase, amplitude, truncation, apodization, and some other distortions involved in the generalized FID-models (1, 2).
References:
[1] Wouters J. M., Petersson G. A.: "Reference Lineshape Adjusted Difference Spectroscopy. I. Theory, II. Experimental Verification. J. Magn. Reson. 28 (1977), pp. 81 - 91 and 93 - 104
[2] Taquin J.: "Line-shape and resolution enhancement of high-resolution F.T.N.M.R. in an inhomogeneous magnetic field". Revue de Physique Apliquee 14, 1979, pp. 669 - 681.
[3] Chmurny G. N. - Hoult D.: "The Ancient and Honourable Art of Shimming". Concepts in Magnetic Resonance, 1990, 2, 131 - 149
[4] Malczyk R., Gottvald A.: "Modelling Inhomogeneity Phenomena in MRS". In: Proc. of the 13th International BIOSIGNAL'96 Conference, Brno (CR), 1996, pp. 95-97
Fig. 1: An Exponential-Gaussian type of an inhomogeneity function (upper figure) and a corresponding inhomogeneity-distorted spectrum with an additive Gaussian white noise (lower figure).
Fig. 2: A periodic spin-echo inhomogeneity function (upper figure) and a corresponding inhomogeneity-distorted spectrum (lower figure). The spin echo experiment corresponds to the inhomogeneity function in Fig. 1. Note a resolution enhancement and a substantial signal-to-noise improvement. The same 3-line spectrum is hidden in both Fig. 1 and Fig. 2 (see a small inserted picture in Fig. 1).